Brownian Motion for Quant Finance
🚀 Master Quantitative Skills with Quant Guild https://quantguild.com 📅 Meet with me 1:1 https://calendly.com/quantguild-support 📈 Interactive Brokers for Algorithmic Trading https://www.interactivebrokers.com/mk... 👾 Join the Quant Guild Discord server here / discord ___________________________________________ 🪐 Jupyter Notebook https://github.com/romanmichaelpaoluc... *TL;DW Executive Summary* Random variables represent a set of possible outomces with associated probabilities, we can *NEVER* predict the outcome of any one event Random variables have statistical and distribution convergence but in reality we are modeling uncertainty which lacks convergence by non-stationarity Normal random variables are useful as sample averages by the CLT follow a normal distribution and are useful for bridging probability and statistics Brownian motion is also defined in terms of a normal (Gaussian) distribution, quite an important distribution! Stochastic processes are a collection of random variables indexed by a draw or even time, we can *NEVER* predict them either Brownian motion is defined by a zero mean, independent and stationary Gaussian increments, and is a.s. continuous We observe how uncertainty increases in the increment the larger the time step capture time varying dynamics, the core use of stochastic processes In practice, we may choose to model a stock price process with a stochastic process using Brownian motion to model the uncertainty We see even in stochastic processes we observe the nice properties of convergence allowing us to easily simulate option prices In reality, more sophisticated modeling is required (Black-Scholes, and beyond for modeling skew/term structure of volatility to extrapolate prices) I hope you enjoyed! Roman ___________________________________________ 📖 Chapters: 00:00 - Bridging the Gap Between Theory and Practice 03:11 - Random Variables (PDFs, CDFs, CFs) 04:34 - Population (Data Generating) and Empirical Distributions 07:07 - Example: Normal (Gaussian) Random Variable 10:02 - Random Variable Statistics: Mean, Variance 12:35 - Law of Large Numbers (LLN) and Statistical Convergence 14:47 - Probability and Distribution Convergence (Bernoulli's) 16:20 - Randomness vs Uncertainty (Theory to Practice) 18:34 - Stochastic Processes 21:35 - Convergence of Stochastic Processes (LLN) 23:22 - Brownian Motion (Theory) 29:02 - Brownian Motion (Practice) 31:32 - Pricing European Option Contracts 35:25 - Roman, Your Assumptions are Wrong 36:26 - TL;DW Executive Summary ___________________________________________ 🗣️ Shout Outs A special thank you to my members on YouTube for supporting my channel and enabling me to continue to create videos just like this one! ⭐ Quant Guild Directors Dr. Jason Pirozzolo ___________________________________________ ▶️ Related Videos Referenced Videos 👉 Markov Chains for Quant Finance • Markov Chains for Quant Finance Master Volatility with ARCH & GARCH Models • Master Volatility with ARCH & GARCH Models Quant Builds 🔨 How to Build a Volatility Trading Dashboard in Python with Interactive Brokers • How to Build a Volatility Trading Dashboar... Statistics and Trading Profitability Over Time (Edge) 📈 Expected Stock Returns Don't Exist • Expected Stock Returns Don't Exist How to Trade • How to Trade How to Trade Option Implied Volatility • How to Trade Option Implied Volatility How to Trade with an Edge • How to Trade with an Edge Quant on Trading and Investing • Quant on Trading and Investing ___________________________________________ 🗂️ Resources 📚 Quant Guild Library: https://github.com/romanmichaelpaoluc... 🌎 GitHub: https://github.com/RomanMichaelPaolucci https://github.com/Quant-Guild 📝 Medium (Blog): / quantguild / quant ___________________________________________ 🛠️ Projects The Gaussian Cookbook: https://gaussiancookbook.com Recipes for simulating stochastic processes: https://papers.ssrn.com/sol3/papers.c... ___________________________________________ 💬 Socials TikTok: / quantguild Instagram: / quantguild X/Twitter: https://x.com/quantguild/ LinkedIn (personal): / rmp99 LinkedIn (company): / quant-guild ___________________________________________
